The method of multiple timescales is widely used in engineering and mathematical physics. In this article we draw attention to the literature on the comparison of various perturbation methods. We indicate where we can obtain an advantage from the concept of timescales, and we present examples where the anticipation of timescales makes sense and cases where, because of resonances or bifurcations, the analysis is less straightforward. In a number of problems second order approximations are

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... al to understand the phenomena. We will conclude from these examples that the anticipation of timescales as in the multiple timescales method may give misleading results; methods that do not anticipate timescales a priori such as averaging and renormalization should be used in research problems. 255 256 FERDINAND VERHULST unbounded with time, so-called secular terms. These secular terms assume different forms and are called timelike variables or timescales. In this elementary problem, the timescales t, εt, and ε 2 t all play a part. Example 2. A well-known example is the damped harmonic oscillator Usually one chooses μ rather small to avoid quenching the oscillation too quickly. Suppose now that we are considering a mechanical process where, for some reason, the damping slowly increases from (say) μ = ε to μ = 2ε. For this oscillator, we propose the equationẍ Note that in the equation a timescale, εt, is already present, but maybe the dynamics of this oscillator will produce more timescales. If t = 0, we have the damped oscillator given above for μ = ε; if we let t tend to infinity, we have this oscillator with μ = 2ε. What happens for the time in between? If ε = 0, the independent variable is time t. It is natural to assume that as the damping varies with εt, an approximation of the problem can be achieved by assuming that two timescales play a part: t and εt. We will show how to handle such a problem. The picture of timescales is not always so simple as in the examples above, and there are many other types of perturbation problems. Consider, for instance, the following example of the classical Euler equation. Example 3.